This used to be about my music Now it is just overflow from other places.

Saturday, July 23, 2016

Circling around the semantic underpinnings of symbolic logic

It is a goal of mine to better understand how mathematical necessity comes about. It some cases it seems to come from cultural games that develop expectations (like counting). Others arise naturally in the vocabulary of boxes and containers and, generally, 'thing' and 'place' words. But above all, I would hope to derive the necessity of symbolic logic ideas from simpler semantics. That hasn't happened yet but I am getting some insights into how the proto semantic operators are related to language usage in comparison with how logic operators co-opt the same usages.

The ',' of proto semantics is equivalent to the natural usage of the word "then", as in "we went to town, then we came home, then we ate lunch". Pretty close to "we went to town and we came home, and we ate lunch". Now this same "and" was borrowed by symbolic logic to mean a version of sequence that is independent of order. To question whether order is important in a sequential statement is to create a straw horse, meaningful to the logicians' co-opted version of "and" but which is not part of the original natural usage of "and".

Very much the same is true for "or" which does not really have a representation in proto semantics. It means, generally, to make a choice. To ask about whether it is an "exclusive" choice creates a straw horse, meaningful to the logicians who co-opted the term but not part of its original natural usage. A choice is a choice and the additional ("and not both") is an artificial add-on from logical usage. So why is "or" not part of proto semantics? I suppose it is too close to requiring a concept of 'collection' that is not available in "proto" land.

Finally the '::' of proto semantics is the "because" or "so" of natural language. It's closest analog in logic is the "therefor" of syllogism. But interestingly, it often corresponds with a different natural usage of the word "and" [example?]. "He sailed beyond the horizon and that was the last they saw of him".

Update (in favor of proto semantics): the two natural meanings of "and" are captured by the notations ',' and '::'. Proto semantics has no concept of set, so no concept of "or" and choice - although that might be added in a post-proto semantics. Logicians have added the (unnatural) assumption of order-independence to the definition of "and" and the (unnatural) assumption of "both/not both" to the definition of "or". Meanwhile the natural word "then" is captured as ',' and is not locked into the logicians warm embrace of the "if....then..." format. Also the word "if" is in no way special in proto semantics. We say "if you are not busy after work then you should come over for drinks" and take this as a statement with a ',' in it for the "then". The "if" partakes more a matter of choice availability - something at the level of "or" and a bit out of reach to proto semantics.